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ICSE Class 6  >  Class 6 Notes  >  Mathematics   >  Worksheet Solutions: Three-dimensional Shapes

Worksheet Solutions: Three-dimensional Shapes

A. Multiple Choice Questions

Q1. Which solid has 6 faces (all squares), 12 edges, and 8 vertices?
(a) Cuboid 
(b) Cube 
(c) Cylinder 
(d) Square pyramid
Ans: (b) Cube
Sol: A cube has 6 square faces, 12 edges, 8 vertices. (Cuboid has rectangular faces; cylinder has curved face and no vertices.)

Q2. Which is a 3D object?
(a) A line drawn with a pencil 
(b) A sheet of paper
(c) A shoe box 
(d) A dot on the board
Ans: (c) A shoe box
Sol: Shoe box has length, breadth, height (3D). A line is 1D, paper is 2D, dot is 0D (just location).

Q3. How many faces does a triangular prism have?
(a) 4 
(b) 5 
(c) 6 
(d) 7
Ans: (b) 5
Sol: Two triangular bases + three rectangular faces = 5.

Q4. Which of these is NOT a polyhedron?
(a) Cuboid 
(b) Prism 
(c) Pyramid 
(d) Sphere
Ans: (d) Sphere
Sol: Polyhedra have only flat faces. Sphere has a curved face.

Q5. A cylinder's net consists of:
(a) 1 rectangle only
(b) 2 circles and 1 rectangle
(c) 3 rectangles
(d) 1 circle and 1 triangle
Ans: (b) 2 circles and 1 rectangle
Sol: The rectangle wraps the curved surface; circles are the two bases.

B. Short Answer Questions

Q6. Name faces, edges, and vertices of a triangular prism.
Ans: 5 faces, 9 edges, 6 vertices
Sol: Two triangular faces + three rectangular faces = 5. Edges: 3 on each triangle (6) + 3 joining corresponding vertices (3) = 9. Vertices: 3 on each triangle = 6.

Q7. A cuboid measures 5 cm by 4 cm by 3 cm. How many unit cubes of 1 cm³ can fill it completely?
Ans: 60 cubes
Sol: Number of unit cubes = volume in cubic centimetres = 5 × 4 × 3 = 60.

Q8. For a square pyramid, state the number of faces, edges, and vertices.
Ans: 5 faces, 8 edges, 5 vertices
Sol: Faces: 1 square base + 4 triangles = 5.
Edges: 4 on base + 4 rising to apex = 8.
Vertices: 4 base corners + 1 apex = 5.

Q9. A solid has 6 faces (all rectangles), 12 edges, and 8 vertices. Name the solid and explain.
Ans: Cuboid
Sol: A cuboid's faces are rectangles (opposite faces equal), and it has exactly 12 edges and 8 vertices.

Q10. In the net of a cylinder, the rectangle's length equals the base's circumference. If radius r = 3 cm and height h = 10 cm, find the rectangle's dimensions. Take π = 3.14.
Ans: Length 18.84 cm; Width 10 cm
Sol:

A cylinder's net = 2 circles (bases) + 1 rectangle (curved surface).

The length of the rectangle = circumference of the circular base.
Formula: Circumference = 2 × π × r.

Substitute r = 3 cm, π = 3.14:
2 × 3.14 × 3 = 18.84 cm.

The width of the rectangle = height of the cylinder = 10 cm.

So, rectangle's dimensions = 18.84 cm × 10 cm.

C. Long Answer Questions

Q11 A gift box is a cuboid of length 12 cm, breadth 8 cm, height 5 cm.
(a) How many 1 cm³ cubes fill it?
(b) What area of wrapping paper is needed to cover it completely (ignore overlap), by thinking of the net?

Ans:
(a) 480 cubes
(b) 392 cm²

Sol (stepwise):

  1. Number of 1 cm³ cubes = volume = l × b × h = 12 × 8 × 5 = 480.

  2. Net of cuboid has 6 rectangles:

    • Two faces of size 12 × 8

    • Two faces of size 8 × 5

    • Two faces of size 12 × 5

  3. Areas:

    • 2 × (12 × 8) = 2 × 96 = 192

    • 2 × (8 × 5) = 2 × 40 = 80

    • 2 × (12 × 5) = 2 × 60 = 120

  4. Total wrapping paper = 192 + 80 + 120 = 392 cm².

Q12. A juice can is a cylinder with radius r = 7 cm and height h = 12 cm. Use π = 22/7.
Find:
(a) Curved Surface Area (CSA)
(b) Total Surface Area (TSA)
(c) Volume

Ans:
(a) CSA = 528 cm²
(b) TSA = 836 cm²
(c) Volume = 1848 cm³

Sol (stepwise):

  1. CSA formula: 2πrh
    = 2 × (22/7) × 7 × 12
    = 2 × 22 × 12
    = 528 cm².

  2. TSA formula: 2πr(r + h)
    = 2 × (22/7) × 7 × (7 + 12)
    = 2 × 22 × 19
    = 836 cm².

  3. Volume formula: πr²h
    = (22/7) × 7 × 7 × 12
    = 22 × 7 × 12
    = 1848 cm³.

Q13. A solid is made by stacking unit cubes into a rectangular block of size: length = 4 cubes, height = 3 cubes, depth = 2 cubes.
(a) How many unit cubes are in the solid?
(b) How many squares appear in the top view (looking from above)?
(c) How many squares appear in the front view (looking from the length side)?
(d) How many squares appear in the side view (looking from the depth side)?

Ans:
(a) 24 cubes
(b) 8 squares
(c) 12 squares
(d) 6 squares

Sol (stepwise):

  1. Total cubes = length × height × depth = 4 × 3 × 2 = 24.

  2. Top view shows length × depth = 4 × 2 = 8 squares.

  3. Front view shows length × height = 4 × 3 = 12 squares.

  4. Side view shows depth × height = 2 × 3 = 6 squares.

The document Worksheet Solutions: Three-dimensional Shapes is a part of the Class 6 Course Mathematics Class 6 ICSE.
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FAQs on Worksheet Solutions: Three-dimensional Shapes

1. What are three-dimensional shapes and how do they differ from two-dimensional shapes?
Ans. Three-dimensional shapes, or 3D shapes, have depth in addition to width and height, making them occupy physical space. Examples include cubes, spheres, and cylinders. In contrast, two-dimensional shapes, or 2D shapes, only have width and height, such as squares, circles, and triangles. The main difference lies in the presence of volume in 3D shapes, which allows them to be measured in cubic units, whereas 2D shapes are measured in square units.
2. Can you name some common examples of three-dimensional shapes and their properties?
Ans. Common examples of three-dimensional shapes include: - Cube: Six equal square faces, twelve edges, and eight vertices. - Sphere: A perfectly round shape with no edges or vertices. - Cylinder: Two circular bases connected by a curved surface, with two edges and no vertices. - Cone: A circular base that tapers to a point, with one edge and one vertex. Each shape has unique properties that dictate how they interact with space and light.
3. How can we calculate the volume of three-dimensional shapes?
Ans. The volume of three-dimensional shapes can be calculated using specific formulas: - Cube: Volume = side³. - Rectangular Prism: Volume = length × width × height. - Sphere: Volume = (4/3)πr³, where r is the radius. - Cylinder: Volume = πr²h, where r is the radius of the base and h is the height. - Cone: Volume = (1/3)πr²h. These formulas help in determining how much space an object occupies.
4. Why are three-dimensional shapes important in real life?
Ans. Three-dimensional shapes are crucial in real life as they are the foundation of architecture, engineering, and design. They enable the creation of structures, vehicles, and everyday objects. Understanding 3D shapes allows for better visualization of space, aids in manufacturing processes, and is essential in fields such as computer graphics and animation, where realistic representations are needed.
5. How do we represent three-dimensional shapes in two-dimensional drawings?
Ans. Three-dimensional shapes can be represented in two-dimensional drawings using techniques such as perspective drawing, isometric projection, and orthographic projection. These methods help depict the depth and volume of 3D objects on flat surfaces. For instance, perspective drawing creates an illusion of depth by converging lines towards a vanishing point, while isometric projection maintains equal angles to represent dimensions accurately without distortion.
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